In order to understand and study this system analytically in detail, we will introduce a further simplifying assumption,
and focus on models with a preferred spatial direction, which are known as locally rotationally symmetric (LRS) spacetimesThese spacetimes have been largely studied in cosmology; check, e.g.,  [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}.. In this case, by choosing the third direction as the preferred one,
this corresponds to assuming \(a_1=a_2\)  and thus the shape-parameter \(\beta _-\) , as well as its conjugate momentum \(p_-\) ,
are vanishing: \(\beta _-=0=p_-\) . Therefore, defining for compactness \(\beta :=\beta _+\)  and \(p:=p_+\) ,
in this LRS model
the Hamiltonian (REF ) is reduced to
\(H=\big (p^2+2e^{4\alpha }U\big )^{1/2}=\big (p^2+2e^{4\alpha -8\beta }\big )^{1/2}.\) 
